Computational complexity of SAT, XSAT and NAE-SAT for linear and mixed Horn CNF formulas

Zusammenfassung Abstract The Boolean conjunctive normal form (CNF) satisfiability problem, called SAT for short, gets as input a CNF formula and has to decide whether this formula admits a satisfying truth assignment. As is well known, the remarkable result by S. Cook in 1971 established SAT as the first and genuine complete problem for the complexity class NP [15]. Thus SAT resides at the heart of the NP 6= P conjecture of complexity theory. In this thesis we consider SAT for a subclass of CNF, the so called Mixed Horn formula class (MHF). A formula F ∈ MHF consists of a 2-CNF part P and a Horn part H. We propose that MHF has a central relevance in CNF because many prominent NP-complete problems, e.g. Feedback Vertex Set, Vertex Cover, Dominating Set and Hitting Set, can easily be encoded as MHF. Furthermore, we show that SAT remains NP-complete for some interesting subclasses of MHF. We also provide algorithms for some of these subclasses solving SAT in a better running time than O(2) = O(( 3 √ 3)) which is the best bound for MHF so far. One of these subclasses consists of formulas, where the Horn part is negative monotone and the variable graph corresponding to the positive 2-CNF part P consists of disjoint triangles only. Regarding the other subclass consisting of certain k-uniform linear mixed Horn formulas, we provide an algorithm solving SAT in time O(( k √ k)), for k ≥ 4. Additionally, we consider mixed Horn formulas F = P∧H ∈MHF for which holds: H is negative monotone, |c| ≤ 3, for all c ∈ H, and P consists of positive monotone 2-clauses. We solve SAT in running time O(1.325) for this formula class by using the autarky principle, that means we can provide a better running time than the so far best one of O(p(n)· 1.427) by S. Kottler, M. Kaufmann and C. Sinz [29] for this class of mixed Horn formulas. Afterwards we consider mixed Horn formulas F = P ∧ H ∈ MHF for which holds: GP consists of disjoint triangles, edges and isolated vertices and H consists of Horn clauses which have at most three literals, but are not necessarily negative monotone and V (P ) = V (H). We can solve SAT in running time O(1.41) for this formula class by applying the autarky principle. Thereafter we consider some interesting subclasses of MHF for which SAT can be solved in polynomial-time. Furthermore, we present an algorithm which solves SAT for mixed Horn formulas with a linear, negative monotone and k-uniform Horn part and a P part which consists of positive monotone and disjoint 2-clauses only. Experimental results lead to the strong conjecture that its running time is betterThe Boolean conjunctive normal form (CNF) satisfiability problem, called SAT for short, gets as input a CNF formula and has to decide whether this formula admits a satisfying truth assignment. As is well known, the remarkable result by S. Cook in 1971 established SAT as the first and genuine complete problem for the complexity class NP [15]. Thus SAT resides at the heart of the NP 6= P conjecture of complexity theory. In this thesis we consider SAT for a subclass of CNF, the so called Mixed Horn formula class (MHF). A formula F ∈ MHF consists of a 2-CNF part P and a Horn part H. We propose that MHF has a central relevance in CNF because many prominent NP-complete problems, e.g. Feedback Vertex Set, Vertex Cover, Dominating Set and Hitting Set, can easily be encoded as MHF. Furthermore, we show that SAT remains NP-complete for some interesting subclasses of MHF. We also provide algorithms for some of these subclasses solving SAT in a better running time than O(2) = O(( 3 √ 3)) which is the best bound for MHF so far. One of these subclasses consists of formulas, where the Horn part is negative monotone and the variable graph corresponding to the positive 2-CNF part P consists of disjoint triangles only. Regarding the other subclass consisting of certain k-uniform linear mixed Horn formulas, we provide an algorithm solving SAT in time O(( k √ k)), for k ≥ 4. Additionally, we consider mixed Horn formulas F = P∧H ∈MHF for which holds: H is negative monotone, |c| ≤ 3, for all c ∈ H, and P consists of positive monotone 2-clauses. We solve SAT in running time O(1.325) for this formula class by using the autarky principle, that means we can provide a better running time than the so far best one of O(p(n)· 1.427) by S. Kottler, M. Kaufmann and C. Sinz [29] for this class of mixed Horn formulas. Afterwards we consider mixed Horn formulas F = P ∧ H ∈ MHF for which holds: GP consists of disjoint triangles, edges and isolated vertices and H consists of Horn clauses which have at most three literals, but are not necessarily negative monotone and V (P ) = V (H). We can solve SAT in running time O(1.41) for this formula class by applying the autarky principle. Thereafter we consider some interesting subclasses of MHF for which SAT can be solved in polynomial-time. Furthermore, we present an algorithm which solves SAT for mixed Horn formulas with a linear, negative monotone and k-uniform Horn part and a P part which consists of positive monotone and disjoint 2-clauses only. Experimental results lead to the strong conjecture that its running time is better 138 Appendix C. Abstract/Zusammenfassung than O(( 3 √ 3)), where n is the number of variables. In addition, we investigate the computational complexity of some prominent variants of SAT, namely not-all-equal SAT (NAE-SAT) and exact SAT (XSAT) restricted to the class of linear CNF formulas. Clauses of a linear formula pairwise have at most one variable in common. We show that NAE-SAT and XSAT are NP-complete for monotone and linear formulas where clauses have length greater or equal k, k ≥ 3. We also prove the NP-completeness of XSAT for CNF formulas which are l-regular meaning that every variable occurs exactly l times, where l ≥ 3 is a fixed integer. On that basis, we can provide the NP-completeness of XSAT for the subclass of linear and l-regular formulas. This result is transferable to the monotone case. Moreover, we provide an algorithm solving XSAT for the subclass of monotone, linear and l-regular formulas faster than the so far best algorithm from J. M. Byskov et al. for CNF-XSAT with a running time of O(2) [12]. Using some connections to finite projective planes, we can also show that XSAT remains NP-complete for linear and l-regular formulas that in addition are l-uniform whenever l = q + 1, where q is a prime power. Thus XSAT most likely is NP-complete for the other values of l ≥ 3, too. Apart from that, we are interested in exact linear formulas: Here each pair of distinct clauses has exactly one variable in common. We show that NAE-SAT is polynomial-time decidable restricted to exact linear formulas. Reinterpreting this result enables us to give a partial answer to a long-standing open question mentioned by T. Eiter in [20]: Classify the computational complexity of the symmetrical intersecting unsatisfiability problem (SIM-UNSAT). Then we show the NP-completeness of XSAT for monotone and exact linear formulas, which we can also establish for the subclass of formulas whose clauses have length at least k, k ≥ 3. This is somehow surprising since both SAT and not-all-equal SAT are polynomial-time solvable for exact linear formulas [42]. However, for k ∈ {3, 4, 5, 6} we can show that XSAT is polynomial-time solvable for the k-uniform, monotone and exact linear formula class. An additional contribution of this thesis is the investigation of the computational complexity of the counting problem #SAT for k-outerplanar formulas, which in general is #P-complete. A CNF formula is called k-outerplanar if its variable-clause graph has a k-outerplanar embedding. First of all we provide an algorithm solving SAT in linear time for the 1-outerplanar formula class. Thereafter we present an algorithm which also solves #SAT in linear time for a given 1-outerplanar formula, whose graph has either no cycles or consists of disjoint cycles without chords. For 1-outerplanar formulas over n variables whose graphs may have cycles we solve #SAT in time O(n) using the separator theorem of Lipton & Tarjan [35]. More generally, we show that #SAT for k-outerplanar graphs, k > 1, can be solved in time O(n) with this prominent separator theorem of Lipton & Tarjan. For formulas having a k-circular-levelplanar graph we solve #SAT in time O(k· 16· ( 2 3 ) 5.13·log2 n) by the separator theorem of Lipton & Tarjan establishing its fixed-parameter tractability with respect to the parameter k. While we need polynomial-time to solve #SAT for a k-outerplanar formula F using the separator theorem of Lipton and Tarjan, we can actually solve #SAT in linear time using the technique of a nice tree decomposition of width at most 3k− 1 for the variable-clause graph of a k-outerplanar formula, as is introduced by H.L. Bodlaender and T. Kloks in [7]. We present an algorithm which uses dynamic programming bottom-up from the leaves to the root in a nice tree decomposition of width at most 3k − 1 of GF to solve the #SAT problem in linear time for a k-outerplanar formula F . Finally, we are able to solve #SAT for Knuth’s nested